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Proof step 271 from proof int2bytes-rec-pos-any (previous step)

# firstn(n, int2bits(j % 2 ^ (i * 8))) ≤ 8, # int2bits(j % 2 ^ (i * 8)) = (i→n(i) - 1) * 8 + n, 0 ≤ bytes2int(i2b(bits2int(firstn(n, int2bits(j % 2 ^ (i * 8))))) ' + bits2bytes(restn(n, int2bits(j % 2 ^ (i * 8))))), 8 < (i→n(i) - 1) * 8 + n, n < 8 + (i→n(i) - 1) * 8, n < (i→n(i) - 1) * 8 + 7, 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), n < 8, 0 < i ⊦ n = 0, i→n(i) - 1 = 0, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], 1 + 8 * i→n(i) = (i→n(i) - 1) * 8 + n, (i→n(i) - 1) * 8 + n = 8 * i→n(i), i = 0, (i→n(i) - 1) * 8 = 1, bits2bytes(firstn(8, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)))) = i2b(bits2int(firstn(n, int2bits(j % 2 ^ (i * 8))))) '

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 272:
# int2bits(j % 2 ^ (i * 8)) = (i→n(i) - 1) * 8 + n, 8 < (i→n(i) - 1) * 8 + n, n < 8 + (i→n(i) - 1) * 8, n < (i→n(i) - 1) * 8 + 7, 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), n < 8, 0 < i, 0 ≤ bytes2int(i2b(bits2int(firstn(n, int2bits(j % 2 ^ (i * 8))))) ' + bits2bytes(restn(n, int2bits(j % 2 ^ (i * 8))))) ⊦ n = 0, i→n(i) - 1 = 0, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], 1 + 8 * i→n(i) = (i→n(i) - 1) * 8 + n, (i→n(i) - 1) * 8 + n = 8 * i→n(i), i = 0, (i→n(i) - 1) * 8 = 1, bits2bytes(firstn(8, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)))) = i2b(bits2int(firstn(n, int2bits(j % 2 ^ (i * 8))))) '

Rule argument

Additional information

Used simplifier rules: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward)% ⊦ m + n = n + m;% ⊦ m * n = n * m;% ⊦ m < n → m ≤ n;% ⊦ # firstn(n, bits) = n;% ⊦ i * j = j * i;